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LIB R.AFLY
OF THE
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OF ILLI NOIS
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no. 40-49
cop. 2
Digitized by the Internet Archive
in 2013
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ANTENNA LABORATORY
Technical Report No. 46
NEW CIRCULARLY POLARIZED FREQUENCY
INDEPENDENT ANTENNAS WITH CONICAL BEAM
OR OMNIDIRECTIONAL PATTERNS
by
John D. Dyson and Paul E. Mayes
20 June 1960
Contract AF33(616)-6079
Project No. 9-(13-6278) Task 40572
Sponsored by:
WRIGHT AIR DEVELOPMENT CENTER
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois
This report was presented, in condensed form, as a paper at the
URSI-IRE Spring Meeting, Washington, D.C., 3 May 1960, under the
title, "The Log-Spiral Omnidirectional Circularly Polarized
Antenna."
i i 3 \pZ>
- ENGINEERING LIBRARY
ACKNOWLEDGMENT
The authors are pleased to acknowledge the assistance of Professor
G. A. Deschamps in formulating the theory of excitation of multi-arm
antennas. Discussions with W. T. Patton were helpful. 0. L. McClelland
supervised the measurements program.
ABSTRACT
A conical beam may be obtained from balanced equiangular spiral
antennas by constructing an antenna with more than two spiral arms and
symmetrically connecting these arms to provide a suppression of the
radiated fields on the axis of the antenna. The angle of this conical
beam can be controlled and with proper choice of parameters it can be
confined to the immediate vicinity of the azimuthal (0 = 90 ) plane.
An antenna with four symmetrically spaced arms can provide a
radiation pattern that is within 3 db of omnidirectional circularly
polarized coverage. The standing wave ratio of this antenna referred
to a 50 ohm coaxial cable is less than 2 to 1 over the pattern bandwidth,
This four-arm version retains the wide frequency bandwidths of the
basic conical log-spiral antenna, and it provides a coverage which here-
tofore has been difficult to obtain even with narrow band antennas.
CONTENTS
Page
1. Introduction 1
2. The Conical Log-Spiral Antenna 3
2.1 The Basic Structure 3
2.2 The Radiation Pattern Beamwidth 5
3. The Conical Beam Antenna 7
3.1 The Principle 7
3.2 Radiation Patterns 9
3.3 Pattern Beamwidth 17
3.4 The Input Impedance 17
3.5 Operating Bandwidth as a Function of Antenna Size 17
4. A Non-Frequency Independent Version 20
5. Conclusions 24
Appendix 25
ILLUSTRATIONS
Figure
Number Page
1. A conical log-spiral antenna with associated coordinate system 4
2. Variation in electric field pattern of typical balanced 2 arm conical
equiangular spiral antenna 6
3. Possible feeding arrangements for multi-arm structures 8
4. "infinite balun" feed used on a four arm conical beam antenna 10
5. Typical electric field patterns and orientation of the conical beam
as a function of the rate of spiral 11
6. Antenna C - 15 - 9 etched from copper clad teflon impregnated
fiberglass 13
7. Electric field patterns of a balanced symmetrical 4 arm conical
equiangular spiral antenna 14
8. Electric field patterns of a balanced symmetrical 4 arm conical
equiangular spiral antenna 15
9. Azimuthal coverage of the radiation patterns in Figures 7 & 8 16
10. VSWR of typical 4 arm conical equiangular spiral 18
11. Projection of equiangular spiral and Archimedes spiral curves on
a conical surface 21
12. Electric field patterns of symmetrical 4 arm conical antennas 23
1A. Terminal region of a structure having N-fold rotational symmetry 26
2A. Instataneous electric vectors at 9 = and 6=9 for three values
of m ° 30
1. INTRODUCTION
The balanced planar and conical equiangular spiral antennas have been
demonstrated to have essentially frequency independent radiation and impedance
characteristics over bandwidths which are at the discretion of the design
12 *
engineer ' . These antennas, based upon the equiangular or log-spiral curve,
have the property that the highest and lowest usable frequencies are independent
The highest usable frequency is determined by the diameter of the truncated
region at the origin, which must remain small in terms of the operating
wavelength, and the lowest usable frequency by the arm length and hence
the maximum diameter of the antenna.
The two-arm planar antenna provides circularly polarized, single lobe,
bidirectional radiation on the axis of the antenna. An orthogonal projection
of the two-arm planar antenna on a conical surface forms an antenna which,
over a suitable range of parameters, confines the radiation to a single
lobe directed off the apex of the cone.
It is possible to devise a multitude of frequency independent structures
by using the log-spiral arm as a basic element. Many different excitations
may be used when several log-spiral arms with a common origin are placed
on a cone or plane. Some configurations and excitations produce radiation
patterns which are distinctly different from those obtained heretofore.
* A convenient abbreviation for logarithmic spiral, a synonym for equiangular
spiral.
1. J. D. Dyson, "The Equiangular Spiral Antenna," IRE Trans, on Antennas and
Propagation, vol. AP-7, pp. 181-187, April, 1959. Also Technical Report
No. 21, University of Illinois Antenna Laboratory, September 15, 1957.
2. J. D. Dyson, "The Unidirectional Equiangular Spiral Antenna," IRE Trans..
on Antennas and Propagation, Vol. AP-7, October 1959* Also Technical Report
No. 33, University of Illinois, Antenna Laboratory, July 10, 1958.
It is the purpose of this paper to introduce a simple theory relating
the excitation and the radiation fields (insofar as now possible), and to
present data showing the performance of the four-arm conical log-spiral
antenna with one particular excitation.
2. THE CONICAL LOG-SPIRAL ANTENNA
2.1 The Basic Structure
The conical log-spiral antenna with its associated coordinate system
is shown in Figure 1. On a plane surface, the edges of one arm of a
logarithmic spiral antenna may be defined by
'■ ' i aff
P = P e
and
/ a a(<P - $)
Since Tan a =
dPV.d0> . a
/ / ( — - — ) <P
P l =P e tan a
The orthogonal projection of P and P_ on the surface of revolution
1 ^
7T - 9 = G is defined by sin 9
tan a ' r
sin f
a " d -• (r -2) (<P-h
where • -sin 6
2_ 6 . /
tan a
K = e
P is the radius vector from the origin to the truncation of the spiral
at the apex region, and P and p are the radius vectors to the inner and
outer edges of the exponentially expanding arm at a given angle <P. The
angle ° is a constant and of such value that if the curve traced out by
P is rotated about the axis through the angle 6 it will coincide with the
curve traced out by p . The angle a, a constant, is the angle between the
radius vector and a tangent to the log-spiral curve at the point of inter-
section.
Figur* 1 A conical log-spiral antenna with
associated coordinate system
5
The second arm of the balanced structure is defined by rotating curves
1 and 2 through 7T radians. Hence the defining parameters are: the included
cone angle, 26 ; the arm width determined by °, or K; the rate of spiral, a,
the base diameter, D; and the apex diameter, d.
2.2 The Radiation Pattern Beamwidth
The beam width of the two-arm conical antenna can be controlled over
a limited range by a suitable choice in the rate of spiral. Figure 2
shows typical electric field radiation patterns for balanced two arm
.o „„o , ,_o
ant
ennas constructed with an a of 73 60 and 45 . Typical half-power
beamwidths range from 60-70° for a of 82 , 70-80 for a of 73°, and 160-180°
o o
for an a of 60 . As the angle a is decreased to 45 the beamwidth increases
o
to 180-200 . Pattern cuts through the axis and perpendicular to the axis
of the antenna [Figure 2(c)] indicate that this latter case provides essentially
circularly polarized coverage in one hemisphere and omnidirectional coverage
on the 6 = 90 plane.
It has been pointed out that a modified version of the balanced conical
antenna is obtained when the width of the arm is made constant rather than
2
tapered . This form of the antenna is readily constructed of wire or cable.
However, it is only an approximation to the true equiangular spiral structure.
The approximation is good for relatively tightly spiraled antennas, i.e., a
greater than 60 , and results in only minor pattern changes. As the angle
a is decreased to the neighborhood of 45 to 50 marked pattern changes occur
for the wire approximation, including a multilobing of the main beam and
large radiation off the base of the cone. Thus to realize the patterns such
as shown in Figure 2(c) the decrease in the rate of spiral must be accompanied
by an increase in the arm width, i.e. increase in o (or K) . The patterns shown
o
in Figure 2(c) were for an antenna constructed with the parameters; 29 = 20 ,
o
a = 45° K = .75 ( 6 = 94°).
<f> =0°, 9 VAR
(1=73'
<£VAR, e=90 <
^.=60*
b.
«=45 (
Figure 2 Variation in electric field pattern
of typical balanced 2 arm conical
equiangular spiral antenna
E^, Eg polarization.
9 = 10 .
o
3. THE CONICAL BEAM ANTENNA
3.1 The Principle
When using multiple-arm structures the number of choices of feeding
3
systems increases . There are basic excitations of multiterminal antennas
which are simply related to the azimuthal variations of fields of the
- -jirfp
form e J associated with solutions of Maxwell's equations. The parameter
m must be an integer to make the field single-valued.
Excitations of the spiral arms which correspond to each of these
radiation "modes" are readily apparent. The customary excitation of the
two-arm spirals, as shown in Figure 3(a), corresponds to m = + 1 and would
be expected to produce a field which varies primarily as e . For antennas
which are not large compared to the wavelength the lower order terms will
be predominant.
With four-arm structures the number of basic excitations increases.
An excitation corresponding to m = 1 is shown in Figure 3(b) . Hence in
order to obtain the e fields with a four-arm spiral it is necessary to
excite the two pairs of arms with a 90 degree phase shift between them.
With the four-arm structures it is possible to produce fields which
correspond to higher values of m . For example, the excitation shown in
Figure 3(c) corresponds to m = +2 and should produce fields which vary
-i2fP
primarily as e . This concept may be generalized to any number of arms,
N, and a discussion of the more general case is given in the Appendix.
3. G. A. Deschamps, "impedance Properties of Complementary Multiterminal
Planar Structures," Trans. IRE, Special Supplement, Vol. AP-7, Dec, 1959,
p. S371. Also Technical Report No. 43, University of Illinois, Antenna
Laboratory, Nov. 11, 1959.
* First pointed out in Quarterly Report No. 5, Contract AF33(616)-6079,
Antenna Laboratory, University of Illinois, 31 December 1959, pp. 9-11.
+ CT7 (a)
+j
-I + IC=3 (b)
"J
fl
D
fl
Figure 3 Possible feeding arrangements
for multi-arm structures
9
Examination of the solutions of Maxwell's equations in spherical
coordinates shows that values of m different from unity are always accompanied
by a null in the 9 - function (associated Legendre polynomials) along the
polar or 8 = axis. Therefore we expect an excitation which corresponds
to any in / + 1 to produce a conical beam. The excitation of four arms
corresponding to m = 2 as shown in Figure 3(c) , is the simplest case.
This lowest order conical beam excitation is readily achieved by
connecting opposite arms together and feeding one pair against the other,
i.e. 180 degrees out of phase. It is apparent also from the symmetry of
the input currents in this case that there will be zero field along the
antenna axis. The antenna can be fed by a balanced feed line, or a coaxial
line and balun, placed on the axis of symmetry. It may also be fed by
carrying the feed cable along one of the arms as outlined in the previous
2
paper m Details of this latter method are shown in Figure 4. The balance
and symmetry of the feed is important if symmetrical patterns are desired.
3.2 Radiation Patterns
Figure 5 shows typical radiation patterns of symmetrical four arm
antennas fed in the manner shown in Figure 3(c). As indicated, the rate
of spiral, (the parameter a), which was the primary factor in controlling
the beamwidth of the balanced two arm antenna, determines the orientation
of the conical beam of the balanced four arm antenna. Conical antennas may
be constructed to provide a conical beam with any angle of orientation
from around 40 to more than 90 off the axis of the antenna. The case
where the beam maximum is located at 9 = 90 is of particular interest
since it fills a need for a simple, very broad band, circularly polarized,
omnidirectional source.
10
\
\
TOP
\
V
SIDE
Figure 4 'infinite balun" feed used
on a four arm conical
beam antenna
<*=73°
oC = 60°
11
Q^=45°
uj
UJ
o
CD
40 r
50 -
60
70 -
- 80
90
/'
•J-
40 50 60 70
o< IN DEGREES
80 90
Figure 5 Typical electric field patterns and
orientation of the conical beam as a
function of the rate of spiral (7.5 ^ < 10 )
• ■
.
Figure 6 shows a typical four arm balanced equiangular spiral antenna
constructed on a 15 cone. This antenna was etched from a flexible, copper-
clad, teflon-impregnated, fiberglass material and then formed into a cone.
The feed cable is rg 141/U. The energized cable is carried along one arm;
dummy cables are placed on the other arms to maintain structural symmetry.
To obtain the desired bandwidth, the arms on this particular antenna were
later extended to a cone base diameter of 31 centimeters. Radiation
patterns of this antenna are shown in Figures 7 and 8 from 550 mc where the
base is .57 wavelengths in diameter up to 4000 mc where the diameter of
truncated apex is approximately 0.2 wavelengths. The patterns are for E~
and E.~ polarized fields. The first two columns are pattern cuts through
the axis of the antenna and the third column is for a cut perpendicular to
the axis, on the U = 90 plane.
The azimuthal coverage shown in these patterns may be examined in greater
detail in Figure 9 where the total deviation in decibels from omnidirectional
coverage is plotted for the orthogonal polarizations. The axial ratio
on the 6 = 90 plane varies somewhat with the angle i P. It is shown for
one particular angle, which is a representative angle of orientation and
not an optimum case. Over a considerable bandwidth the total amplitude
deviation is less than 3 db and the axial ratio is 3 db or less.
Two pattern characteristics should be noted. For large a, the conical
beam patterns are smooth and well formed and, if desired, the arms may be
approximated by wire or cable. As the angle a decreases beyond 60° the
on patterns exhibit minor irregularities and are not as symmetrical.
In ;i ■« loosely spiraled antennas require wider exponentially
expanding arms. Th< ••-.<■ ' h. istics correspond to those noted for the
13
Figure 6 Antenna C - 15 - 9 etched from copper clad
teflon impregnated fiberglass
a = 45° D ■ 20.5 cm
14
8°, OVcuu.
F=550Mc
700 Mc
CJ>-90° e vcuo.
(j)Vojo Q = 9Q <
900 Mc
N \
S \
/
r
^
1200 Mc
Figure 7 Electric field patterns of a
balanced symmetrical 4 arm conical
ilangular spiral antenna.
.", a 1 .", K = .925, D = 31 cm, d = 1.5 cm
0s 0° e "z^t/.
1600 Mc
^=90° e vcuu.
(£ VoMj. ^ Q r 9QO 15
3000 Mc
^
"h
'y.c. .^'^^s-:^
V-< ' -v
7 ^?r
4000 Mc
y
s;
^r=—
\
n
I :
\ '
/
I /
-^~
e =
Figure 8 Electric field patterns of a
balanced symmetrical 4 arm conical
equiangular spiral antenna.
7.5°, a = 45°, K = .925, D = 31 cm, d = 1. 5 cm
a
o
•H
+->
cS
•H
•o
<D
X! 00
-P
eg
«H
O «>
ID (0
bD <D
Q) U>
> 'H
O Ek
o
c
•H
(0
c
Fh
0)
-p
+->
a
a
X!
■p
S
•H
N
a)
h
•H
o08l = 4>
o06=e
OllVti 1VIXV
qp Nl NfcGlJMd
Nonviavd o06=e jo
N0I1VIA3Q 3anilldkNV
17
two arm axial beam antennas.
3.3 Pattern Beamwidth
The beamwidth in a <P = constant plane is relatively insensitive to a
change in antenna parameters. Antennas constructed with both 15 and 20
included cone angles (2« ) with 45 < a < 73 and with cable arms or with
o - -
exponentially expanding arms had half-power beamwidths ranging from 35 to 55
degrees, with an average value of 45 degrees.
3.4 The Input Impedance
The input impedance of the four arm antenna, fed in the manner of
Figure 3(c), rapidly converges to a characteristic value. Antennas constructed
o o
with 15 or 20 included cone angles with RG 8/U arms or exponentially
expanding arms fed with RG 141/U, typically have an input impedance of
from 45 to 55 ohms for a ranging from 45 to 60 degrees. As a is increased
to 73 degrees the impedance rises to the neighborhood of 70 ohms. These
values are approximately one half those noted for similar two arm antennas.
The input voltage standing wave ratio of the antenna referred to in
Figures 7 and 8 is plotted in Figure 10. Note that it is less than 1.5
to 1 referred to 50 ohms over most of the usable pattern bandwidth.
3.5 Operating Bandwidth as a Function of Antenna Size
The usable antenna bandwidth is fundamentally determined by the diameter
of the truncated apex and the antenna arm length. As with the two arm
antennas, the radiation patterns tend to deteriorate as the apex region
approaches 1/4 wavelength in diameter. It was previously noted that the
balanced two arm antenna constructed on a 15 or 20 degree (total apex angle)
cone, with an a of 73 could be operated to a frequency such that the cone
base diameter is on the order of 1/3 wavelength. As the rate of spiral
ed
U
•H
C
o
o
E
u
a
•<# u
•H
h a
O
•H U
a eS
>> rH
tuD
<H C
rt
•H
^ cr
to <D
>
s-t
bO
«
e
o
ro
CD
in
<*
ll
U~ OS 01 Q3dd3J3d tiMSA
19
is decreased, i.e. a decreased to 45 there is not sufficient radiation
surface on this size cone to dissipate the energy without back radiation
and hence the size of the cone must be increased to the order of .6
wavelength at the lowest frequency of operation. The four arm structures
exhibit very similar characteristics and hence omnidirectional coverage on
the = 90 plane requires an antenna whose base diameter is on the order
of .6 to 2/3 wavelength at the lowest operating frequency.
20
4. A NON-FREQUENCY INDEPENDENT VERSION
Thus far we have considered only antennas constructed from the
equiangular spiral curve. These antennas are frequency independent
in the sense that, within the limits imposed by the physical size, the
4
scaling principle is fulfilled . The pattern characteristics of these
log-spiral antennas (such as the beamwidth of the two arm antennas and the
angle of orientation of the conical beam of the four arm antennas) are
constant for a change in the frequency of operation. These characteristics
are directly related to the constant parameter a, which indicates the rate
of spiraling of the arms.
It is possible, as shown in Figure 11, to construct conical antennas
from other curves, such as the Archimedes spiral. Although these antennas
may be operated over wide frequency bands, they are not frequency independent
since the parameter a at any point on the curve is directly related to the
angle <P at that point. As the frequency of operation is changed the active
aperture of the antenna is composed of a structure with a changing rate
of spiral. This shows up as a definite widening of the beamwidth of the
two arm conical Archimedes spiral antenna as the operating frequency is
increased. There is also a variation in the angle of orientation of the
conical beam o! the four arm conical Archimedes spiral antenna with a
change in frequency.
A. V. H. Kurmey, Frequency Independent Antennas," 1957, IRE National
"1, P'. 1, pp. 114-118. Also Technical Report No. 20,
University '-f tlllnoie, Antenna Laboratory, October 25, 1957.
p =ke
Q(j>
21
a = arctan tt
(a)
p = ka<£
<< = arctan <£
(b)
Figure 11 Projection of equiangular spiral
and Archimedes spiral curves on
a conical surface.
22
Radiation patterns for a four arm conical Archimedes spiral antenna
are shown in Figure 12(a) . This antenna was constructed to provide a range
o o
of a from approximately 45 at the apex region to 85 at the base. As
indicated in Figure 12 the complete range of beam orientation from
o o
approximately 45 to 90 off axis is swept out as the frequency is varied
from 1000 to 2000 mc . For comparison, patterns for an equiangular spiral
antenna wound on the same cone are shown in Figure 12(b) .
ARCHIMEDES SPIRAL
P =1.026*
f = 1000 Mc
E *
B
EQUIANGULAR SPIRAL
(
/>= e
sin 10°
tan45°
)4>
23
f = 1400 Mc
f = 2000 Mc
Figure 12 Electric field patterns of
symmetrical 4 arm conical
antennas
© = 10°, D = 29.5 cm, d = 4.5 cm
o ' '
(<P = 90°, Q var pattern)
24
5. CONCLUSIONS
It is possible to obtain a conical beam from the balanced equiangular
spiral antenna by constructing the antenna with more than two arms and
connecting these arms to provide a higher order <P - variation which is
accompanied by an axial null. The angle of this conical beam can be
controlled; in particular, it can be placed in the immediate
o
vicinity of the 6 = 90 plane to provide an omnidirectional pattern.
An antenna with four symmetrically spaced arms can provide a radiation
pattern that is within 3 db of omnidirectional coverage. The standing
wave ratio of this antenna, referred to a 50 ohm coaxial cable, is usually
less than 1.5 to 1 over the pattern bandwidth.
This form of log-spiral retains the extremely wide frequency band-
width and circular polarization properties of the basic conical log-
spiral antenna and it provides a coverage which heretofore has been difficult
to obtain even with narrow band antennas.
Since the Archimedian spiral approximates a log-spiral with changing
parameters, this non-frequency-independent version of the antenna may be
constructed such that the conical beam may be frequency scanned in the
polar angle.
25
APPENDIX
Consider a radiating system composed of a number of identical conductors
on the surface of a cone. Assume that the structure has N-fold rotational
2ir
symmetry, i.e., a rotation about the cone axis through the angle will
leave the structure unchanged. Figures 3(b) and 3(c) of the text show
examples of four-fold symmetry.
The excitation of the conductors will be accomplished at the apex of
the cone where the dimensions are small compared to the wavelength. There
will be N terminals available in the excitation region, symmetrically
spaced about a circle of small radius. Denoting the input current at the
n terminal by I , the excitation can be described by the current vector
I = ( V V v v w
Any index is defined modulo N. To satisfy the requirement of conservation
of current, we note that
N
£ 1=0 (1A)
T n
n = 1
This viewpoint that each possible excitation is represented by a
vector, I leads to the examination of the possible basis for the vectors
of this space. Because of the n-fold symmetry the choice of "symmetrical
components" as base vectors proves more convenient than others.
Rotation of the excitation by one step
: a = ( v v v V (2A)
26
®
/
©
©
©
©
Figure 1A Terminal region of a
structure having N - fold
rotational symmetry
27
2 IT
would simply produce a field which is rotated in space by the angle "
Hence the transformation of excitation
I 1 as PI (3A)
where P is th*~ n x n permutation matrix
1 ...
1 ...
P= 0001. ..00 (4A)
1
1
produces a simple change in the field. The matrix P is a special case
. 27Tk x
of circulant matrices. The eigenvalues of P are exp. ( — ) and the
eigenvectors are
27TR . 27Tk(N-2) . 27Tk(N-l)
-1/2. J N~ J N J N .
A k = N (1, e , , e , e )
(5A)
From these eigenvectors we can obtain the basis of our vector space.
The eigenvectors of P for the case N = 4 are
A 1 = 1/2 (1, j, -1, -j)
A 2 = 1/2 (1, -1 ; 1, -1)
A 3 = 1/2 (1, -j- -1, j)
(6A)
A 4 = 1/2 (1, 1, 1, 1)
Note that A A and A satisfy the condition of Equation (1A) whereas
A does not. The former three vectors provide an orthonormal basis which
spans the vector space of all possible excitations when N = 4.
28
A . A * = 1/4 (1 + 1 + 1 + 1) = 1
A 2 ' A 2 = 1/4 (1 + 1 + 1 + 1) = 1
A 3 . A * = 1/4 (1 + 1 + 1 + 1) = 1
A . A * = 1/4 (1 - j - 1 + j) « (7A)
A . A * = 1/4 (1 - 1 + 1 - 1) =
A 2 . A 3 * = 1/4 (1 - j - 1 + j) =
Let us now examine the properties of these basic excitations.
Excitation I = A produces field in which a rotation by 77/2 is equivalent
to a 7T/2 change in phase. Solutions of Maxwell's equations in spherical
coordinates can be written with azimuthal variation of the form e (m integer)
Of these solutions only those with m = 4k + 1 (k = 0, + 1, + 2, +3...)
will satisfy the above relationship between rotation and change in phase.
Hence the excitation I = A will produce fields which can be expressed
in the following way
F, = L a (P
k 4k + 1 (8A)
where <Pm = g (r 9) e J
in ^
nfP
Similarly, I = A yields m = 4 k + 2 and I = A yields m = 4 k + 3,
^ 3
and these each will produce fields
F = E a <P
P k 4k + p (9A)
A Bimil.ir ftrgtUMnl Cftl i <«d through for arbitrary N.
Ari satisfying (1A) can be expressed as a linear combination
A ] " A 2' A 3 '' Produced thereby as a linear combination of
F ji y y T, " 15i '><>>> de«( ribed by E will produce the fields
29
F = E (A * E*) F (10A)
J J
If the coefficients a above were known, the fields produced by any
excitation would be completely determined. Unf ortunately, this problem
has not yet been solved for log-spiral elements.
Some useful observations can be made for the log-spiral however,
by interpreting the above results in terms of experimental data.
Consider the case N = 2 which has been extensively investigated. The
eigenvectors for the permutation matrix in this case are
\ " \ (1 > ^
A 2 = 1(1,D
(11A)
and only A satisfies Equation (1A) . This corresponds to the excitation
of the two-arm spiral sketched in Figure 3(a) . The field F for the case
N = 2 would be of the form
F = £ a (P
Ik T 2k + 1 (12A)
and the possible ^-variations are
e j(2k + 1) 9
It has been observed that the radiation produced by 2-arm log-spiral
antennas is very nearly circularly-polarized over the major portion of
the beam. Circular polarization with non-zero field on the axis requires
that E^ and Eg vary as cos(<»>t + <f) and sin (<*>t + <P) respectively, the m = 1
case. Functions with higher values of m would contribute to the off-axis
fields. Figure 2A shows a sketch of how the circularly polarized fields
= 0,
m=2
m = 3
7.h Instantaneous electric vectors at
■ and ■ for three values of m.
31
would appear instantaneously around the polar axis for a few of the small
values of m. The values m = 1 and 3 would predominate for the two-arm
case. Note that these add alternately in and out of phase in orthogonal
cross-sections. Hence the rotational symmetry of the beam should provide
an indication of the relative magnitude of these two fields, neglecting all
others. A tightly- wound spiral produces more nearly rotationally-symmetric
beams, hence predominantly the m = 1 case.
The excitation of Figure 3(c) used to produce the conical beams is one
of the eigenvectors of (6A), namely A . This excitation produces fields
F = £ a, <P
2 k 4k + 2
and the experimental results indicate that, for certain parameter choices,
the lowest order terms of this series predominate
F ~ a <P n + a <P „
2 ~ o 2 -1-2
These two cases seem equally likely on the basis of the order of the
functions involved. However these two cases correspond to circular
polarized fields of opposite sense. One sense of polarization is favored
over the other by the direction of the spiral winding. Hence it is reasonable
that one of the coefficients a . a be small, a conjecture which is again
o' -1
confirmed experimentally.